Bijections for faces of braid-type arrangements
Abstract
We establish a general bijective framework for encoding faces of some classical hyperplane arrangements. Precisely, we consider hyperplane arrangements in $\mathbb{R}^n$ whose hyperplanes are all of the form $\{x_i-x_j=s\}$ for some $i,j\in[n]$ and $s\in \mathbb{Z}$. Such an arrangement $A$ is \emph{strongly transitive} if it satisfies the following condition: if $\{x_i-x_j=s\}\notin A$ and $\{x_j-x_k=t\}\notin A$ for some $i,j,k\in [n]$ and $s,t\geq 0$, then $\{x_i-x_k=s+t\}\notin A$. For any strongly transitive arrangement $A$, we establish a bijection between the faces of $A$ and some set of decorated plane trees.
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Advanced Algebra and Geometry
Marriage and Sexual Relationships
Publication details
- Journal / source
- European Journal of Combinatorics
- Publisher
- Elsevier BV
- Volume
- 137
- Pages
- 104410–104410
- Publication date
- 3 June 2026
- Language
- EN